The Kinematics of Jumping of Globular

Seiichi Sudo a*, Masahiro Shiono a, Toshiya Kainuma a, Atsushi Shirai b, and Toshiyuki Hayase b

a Faculty of Systems Science and Technology, Akita Prefectural University, Japan b Institute of Fluid Science, Tohoku University, Japan

Abstract—This paper describes jumping behaviour of have been conducted. For example, studies of the the globular . Direct observations on the energetics and a model of the mechanism in the jump of jumping and climbing behaviour of springtails were the rabbit flea [2] and the locust [3] were reported. The conducted in the laboratory. The kinematics of jumping effect of muscle properties, leg design and jumping in tiny springtails was analyzed with high-speed video technique for humans, other vertebrates and camera system. The vertical climbing behaviour was were investigated [4]. The structure of the hind limbs analyzed in contrast to the jumping behaviour. The and the kinematics of their movements that propel jumping performance of springtails was revealed. jumping in plant hopper insects were analyzed [5]. The jumping behaviour in springtails was investigated by Index Terms— Springtail Jump, Leaping Organ, high-speed photography methods [6,7]. Furcula, Aerodynamic Drag In spite of the number of investigations, however, there still remains a wide, unexplored domain. Research data on morphological characteristics of leaping organ I. INTRODUCTION of globular springtails and the details of jump mechanisms are scarce, and there are many points Most are capable of movement, and they which must be clarified. move about using legs, wings, or fins. The dynamics of In this study, the jumping and climbing these locomotion of animals are a fascinating subject characteristics of a globular springtail were examined that has attracted the attention of biologists, engineers, using high speed video camera system. mathematicians, and other scientific workers for a period of many years. Arthropoda is the largest , and is one of tremendous diversity. The name Arthropoda means II. EXPERIMENTAL METHOD “joint-footed”, and this feature of the , along A schematic diagram of the experimental apparatus with the segmented body structure and the tough outer to study jumping behaviour of globular springtail is skin (cuticular exoskeleton), is a distinctive shown in Fig.1. The experimental apparatus consists of characteristics of the phylum. They are found almost the jumping horizontal plate, the optical measurement everywhere and make up more than three-quarters of all system, and the analysis system. the living organisms on this planet. The jumping plate made of the with dimension In locomotion such as swimming, walking, running, 60 mm  290 mm wide and 10 mm in thickness. In and jumping, organisms with high Reynolds number to guarantee the two-dimensional plane in the tend to be larger and faster moving organisms. The high-speed images, experiments on springtail jump were Reynolds number represents the ratio of inertial force to carried out in the region of 10 mm width space. The viscous force in the flow. For small Reynolds number, optical measurement system is composed of a high- the flow will always be laminar, and viscosity is the speed video camera (Photron FASTCAM-ultima-SE), a resistance of a fluid to flow under the influence of an control unit, a video cassette recorder (SVO260), a applied external force. video monitor (PVM-1450), and a personal computer Springtails are minute insects without wings. (Endeavor Pro 2000). Micro Nikkor lens (55mm f/2.8S) Especially, globular springtails have potato-shaped with the close-up ring was used in the photographing of bodies, usually less than about 1 mm long, and 6 short springtail movements. First, the scales were recorded in legs. They are able to perform remarkable jumps into the view field of the camera. The view field for the air. For such springtail jump, viscosity tries to stop recording the whole of the jumping behavior was set in it. In general, locomotion is one of the major energetic the region of 155mm  155mm. Observations of the costs faced by animals and various strategies have early stages of springtail jump was conducted in the evolved to reduce its cost [1]. Therefore, extensive region of 6mm  3.5mm. In addition, the view field was investigations on the jumping performance of insects modified appropriately.

* Corresponding author: Ebinokuchi 84-4, Tuchiya, Yurihonjo, Akita 015-0055, Japan, E-mail:[email protected]

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Figure 2 shows a photograph of the globular springtail used in this experiment. The head bears a pair of antennae, a pair of eyes, and the mouthpart. The thorax consists of three segments. The abdominal segments fuse making the intersegmental boundaries difficult to resolved as shown in Fig.2.

III. SPRINGTAILS

A. Collembora Springtails belong to a group of insects known as

Collembora [8]. Springtails are primitive, minute, Fig. 1. Experimental apparatus for free jumping analysis of a globular wingless insects. They are widely distributed, and are springtail. found throughout the year inhabiting moist situations. Approximately 2,000 different species of Head Thorax springtails have been identified worldwide. They vary in body shape from elongate to globular. They range in Eye color from white, gray, or yellow to red, orange, purple, brown, or mottled hues. Their have only 4-6 segments, less than any other . Another unusual feature is a tube protruding from the abdomen. The name springtail comes from a forked structure attached to the underside of the abdomen. When this mechanism 1.0 mm is released, the abdominal extension snaps back, tossing Fore leg the springtail into the air. In this study, the globular springtail as shown in Fig.2 was examined. Middle leg Hind leg

Fig. 2. Photograph of a globular springtail used in the jumping B. Vertical Climbing experiment. Many species of springtails have well developed jumping and climbing behaviour [7]. The effects of variation in body form on the mechanics of were examined using the miniature force platform [9]. Humans, bushbabies, frogs, locusts, fleas jump by rapidly extending a pair of legs [4]. In springtail jump, however, a spring-like (furcula) is used. This unusual locomotor organ is independent of three pairs of legs. In this paragraph, the Fig. 3. Video sequence of the springtail climbing. climbing behaviour of springtail was examined. Figure 3 displays a sequence of pictures for the The globular springtail was released at the position climbing behaviour of globular springtail. The pictures on the plate. Free jumping of the springtail on the plate show springtail locomotion formed by the legs. The was recorded with high-speed video camera system, time between successive pictures is t  50 ms. Figure capable of capturing up to 40,500frames per second 4 shows the gait pattern of the globular springtail in (fps); typical rates in this study 4,500-27,000 fps. A climbing of a vertical board. In Fig.4, black bars show series of frames of free jumping behaviour of springtail the swing phase in which a leg lifts and moves forward. were analyzed by the personal computer. The multiple Six characteristic points on the springtail legs were photograph method was used to determine the jumping defined by the signs shown in Fig.4. The insects can trajectory. generate a suitable gait pattern which depends not only The multiple photograph method is a multiple exposure on the walking speed, but also on the external load. In technique on a single picture. The jumping trajectory of general, the insects usually have two kinds of gait the springtail with 1 mm body length was described as a patterns, that is, the tripod gait and the metachronal gait large number of points. Those points give the position [10]. In this experiment, the springtail shows the of the insect with sufficient accuracy in the resolution. complicated gait pattern that the tripod and metachronal Test springtails were collected in the field in gaits were mixed. This fact suggests that the climbing of Yurihonjo, Japan. They were very small (typically 1.0 the vertical board accompanies the heavy load. mm in length), and they have a rounded body shape. Springtails, however, move the long distance by the jump in short time. In Fig.4, xc is the horizontal distance

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In this chapter, jumping kinematics such as the take- off velocity of springtail and the forces acting on the body were considered. In general, when a three- dimensional body is immersed in a fluid and is in relative motion with respect to it, the forces acting on the body are described by the following equations: 1 F   V 2C A (1) L 2 a L

1 F   V 2C A (2) D 2 a D

1 F   V 2C A (3) S 2 a S

where FL is the lift force, FD is the drag force, FS is the

side force, a is the density of fluid (air), V is the

velocity, A is the frontal area, and CL, CD, and CS are the coefficients of lift, drag, and side force respectively. 2 c z If the elastically-stored energy in the springtail’s  2 c

x furcula E is completely converted into kinetic energy, the take-off velocity V of the springtail is described as follows: 1  2E  2 Fig. 4. Gait pattern in vertical climbing. V    (4)    ms  and zc is the vertical distance in the climbing of the springtail. In this experiment, the locomotive velocity in where ms is the mass of the body. The height in a the springtail climbing Vc was obtained as follows: vertical jump HV is given by Eq.(5):

2 2 x z 2 c c 3 (1) E V Vc   2.7 10 m/s H   (5) t V ms g 2g

This value is quite distinct from the jumping velocity Vj, that is, Vj>>Vc. where g is the gravitational acceleration. The time rate In general, many insects are capable of climbing and of change of the linear moment of the insect is equal to walking upside down on diverse substrates using the resultant force acting on the center of mass. If the adhesive structures on their legs [11]. Dynamics of upward force generated by the furcula acts on the center rapid vertical climbing in cockroaches was reported of mass for the time interval t, the velocity V and the [12]. Unfortunately, adhesive mechanism during mean acceleration  is given as follows [16]: climbing of the springtail is not known. F t V  j (6) ms IV. KINEMATIC DESCRIPTION OF SPRINGTAIL JUMP V F V 2    j  (7) Globular springtails are able to jump by a special t m 2 leaping organ (or furcula) as a means of escaping from s predators. Springtails jump by rapidly extending their where Fj is the jumping force, t is the time, and  is the furcula that is normally folded forward under the distance over which the acceleration occurs. abdomen [13]. Jumping is a common means of escaping Assuming that a take-off angle is  against the rapidly from predators or increasing the speed of horizontal ground, then the maximum height H and locomotion [14]. In general, animals use diverse m distance achieved S are given as follows: strategies to reduce the energetically expensive cost of h locomotion [1]. The energy costs of swimming, flying, V 2 sin 2  H  (8) and running related to the animal’s size or weight, that m 2g is, the cost of locomotion increases with the decrease of body weight of animals [15].

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2V 2 cos sin V 2 sin 2 The solution of Eq.(11) for the case of constant Re S   (9) h g g gives the jump height Hs as follows:

 2 2 2  In the analytical model we approximate the springtail 4 Rems 5 a R V (17) H S  ln 1 by a sphere with the same volume as the prolate 2 2 5 a R  8ms g Re  spheroid. In the same manner as the jump analysis in planktonic copepoda [17], the radius of the sphere R is Eq.(17) shows that the height of springtail jump given as follows: depends on the Reynolds number and the mass of springtail. 2  L  R  3   (10)  2  V. EXPERIMENTAL RESULTS AND DISCUSSION ON JUMPING where  is the aspect ratio of springtail body, and L is the body length. The equation of motion of the A. Jumping Movements vertically projected body for the upward-moving phase The sequence of events during a jump of tiny may be written as follows: springtail (L=1.0 mm) is shown in Fig.5 (a) by the multiple photograph method. In this case, the springtail dV (11) jumped at the angle   86 degree to the ground. The ms  madd   FD  ms g  0 dt arrows in Fig.5 (a) show the direction of springtail where madd is the mass of the displaced surrounding fluid (air). We describe ms and madd for the spherical body as follows:

4 m   R3 (12) s 3 s

2 m   R3 (13) add 3 a where s is the density of springtail. In Eq.(11), assuming the motion of the axisymmetric object, FL and FS were ignored. In this paper, the mass 7 of springtail was ms 1.310 kg, therefore madd 11 was madd  6.310 kg. Science ms>>madd, madd can be neglected. The second term on the left-hand side of (a) Whole trajectory (b) Initial part Eq.(11) is the hydrodynamic drag 1 Fig. 5. Jump trajectories of two springtails. F  C  R 2V 2 (14) D 2 D a

The drag coefficient CD depends on the Reynolds number Re:

2RV 2RV Re   (15)   dt dz where  is the coefficient of viscosity and  is the coefficient of kinematic viscosity. As described later, the Reynolds number ranged Re<500 in the springtail z 2 jump. For Allen flow (0.2

5 (16) CD  4 Re Fig. 6. Position, velocity, and acceleration as function of time during a jump.

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the present paper (Movie S1). The change in position, velocity, and acceleration as function of time is shown in Fig.6. It can be seen from Fig.6 that the maximum

height is Hm  0.115 m, the take-off velocity is V  2.0m/s, and the acceleration is d 2 z dt 2 1.8103 m/s2. In this experiment, the mass of springtail was 6 ms  01.310 kg. The height in the vertical jump can be calculated from Eq.(5), that is,

Fig. 7. Three phases in the springtail jump. V 2 2.02 H    0.20m (18) V 2g 29.8

The comparison between the maximum height Hm in Fig.6 and the vertical jump height HV gives HV-Hm=0.08 m. This difference (HV-Hm) is mainly brought by the effect of drag FD acting on the springtail. The jumping force Fj can be calculated from Eq.(7) as follows: Fig. 8. Body rotation of springtail during the jump. d 2 z F  m  0.13106 1.8103  0.23103 N (19) j s dt 2 On the basis of the actions and positions of the body and furcula, three phases in the jump were identified from the frames of high-speed video movies (Fig.7): (i) the take-off phase, which last from the start of furcula movement to the point at which the springtail leaves the ground, (ii) the aerial phase, which lasts from this point to the first touching point of the ground, (iii) the landing phase (or the bouncing phase), measured as the time from the first contact with the ground. At the take-off phase, the springs strikes the ground. The body of the springtail is accelerated through this strike motion. Figure 8 displays a sequence of images for the early stage of springtail jump. The time interval between

Fig. 9. Distance moved by springtail head in the take-off. successive images in Fig.8 is t  0.48 ms. The selected frames from the movie prove the body rotation of springtail during the jump. The direction in body rotation is the backward somersault. Figure 9 shows the details of the take-off phase which was recorded with 27000 fps. In Fig.9, the head point zh was defined as the root of an antenna of the sprigtail. The change of position zh corresponds to the movement during the take-off of springtail. The contact time of the furcula is about t=1.12 ms. Using the contact time, Eq.(19), and Eq.(6), we can calculate the jump velocity as follows: F t 3 3 j 0.2310 1.1210 (20) V   6 1.98m/s ms 0.1310

The value in Eq.(20) agrees the experimental value in Fig.6. In Fig.5 (a), the time required to reach the Fig. 10. Jump trajectory with larger horizontal distance. maximum height ta is ta=0.150 s. On the other hand, the time required to reach the ground from the maximum height t is t =0.161 s. This fact suggests the existence movement, and the symbol  t shows the time interval of d d of locomotion difference between ascending and data plots for jumping motion. Figure 5 (b) shows the descending in the aerial phase of springtail jump. initial stage of jumping with the body rotation of springtail. The video movie of Fig.5 (b) is attached to

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considered that the spring keep their jump under control in the height and direction with spinning. dt dz B. Jumping with larger horizontal distance Figure 10 shows the example of the springtail jump with larger horizontal distance (in the case of Hm

(dz/dt)max =1.65 m/s, and the maximum horizontal velocity is (dx/dt)max=0.55 m/s. In this case, the take-off velocity is V=1.74 m/s. Comparing Fig.6 with Fig.11, it dt dx can be seen that the horizontal jump leads the decrease in the take-off velocity and acceleration.

VI. CONCLUSIONS x 2 2 dt d The jumping analysis of the globular springtail was conducted using the high-speed video camera system. The gait pattern of the springtail in vertical climbing was also examined. The results obtained are summarized as follows:

2 (1) The biological kinematic data of a jumping    dt dz

   globular springtail were obtained. The maximum 

2 3    acceleration of the center of gravity reached 1.8×10 dt dx    m/s2, causing the jump velocity of about 2.0 m/s. (2) The locomotive velocity in the springtail climbing 3

2 was obtained asV  2.710 m/s . This climbing velocity     c z 2 2 dt d

    is very lower than the jumping velocity.  2     x 2 (3) The globular springtail showed the jump with the 2 dt d     high-speed body rotation. The number of revolutions reached 417 times/second.

Fig. 11. Velocity and acceleration as function of time during a jump. ACKNOWLEDGMENT

This work was partly supported by JSPS KAKENHI The drag force FD in Eq.(2) is caused by the flow of air around the springtail. The drag force is a function of Grant Number 22560173. the air velocity and density, frontal area and drag References coefficient. The springtail may be adjusting the drag [1] Glass, A. C., Jorgensen, S. J., Liebsch, N., Sala, J.E., coefficient according to ascending and descending in Norman, B., Hays, G. C., Quintana, F., Grundy, E., the aerial phase. Figure 5 (b) and Fig.8 show body Campagna, C., Trites, A.W., Block, B. A. and Wilson, R. rotation of the springtail in aerial phase. In Fig.3 (b), the P., Convergent Evolution in Locomotory Patterns of Flying and Swimming Animals, Nature Communications, rotation of springtail shows 417 times per second. A DOI: 10.1038(2011), pp.1-7. computer model with the body rotation (spin rate: -8.5 - [2] Bennet-Clark, H. C, and Lucey, E. C. A., The Jump of the 37 revolutions/ second) has been used to analyze jump Flea: A Study of the Energetics and a Model of the kinematics and energetics [13]. However, the springtail Mechanism, Journal of Experimental Biology, Vol.47(1967), pp.59-76. showed higher revolutions in this experiment. Figure 8 [3] Bennet-Clark, H. C., The Energetics of the Jump of the shows the initial stage in springtail jump. It can be seen Locust Schistocerca gregaria, Journal of Experimental from Fig.8 that the springtail of 1 mm body length spins Biology, Vol.63(1975), pp.53-83. through the air. In this state, the springtail spins [4] Alexander. R. McN., Leg Design and Jumping Technique for Humans, Other Vertebrates and Insects, Philosophical backward rapidly through the air. The rotation of the Transactions of the Royal Society B, Vol.347(1995), springtail produces an air circulation around the body. pp.235-248. This fact suggests that globular springtails use the [5] Burrows, M., Jumping Performance of Planthoppers Magnus effect to create lift in Eq.(1). However, the (Hemiptera, Issidae), Journal of Experimental Biology, springtail must be wasted energy by spinning. It can be Vol.212(2009), pp.2844-2855. [6] Christian, E., The Jump of the Springtails, Naturwissenschaften, Vol.65 (1978), pp.495-496.

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